No abstract

This paper is a comprehensive survey of the concepts which generalize first countability, of the relations among the concepts, and of the major examples found in the literature on the topic, 0. Introduction. There are several reasons which one may give to indicate the value of generalizing the first axiom of countability. Among these are the following:(1) To weaken assumptions in important theorems. For example, a closed image of a metric space is metrizable, if the range is assumed to be first countable.(2) To study important properties. For example, if a real valued function f is continuous upon restriction to each compact subspace of a space X, then / is continuous on X, if X is a /c-space.(3) To study sequences and their properties. For example, in first countable spaces every accumulation point of a subset A is the limit of a sequence in A.The emphasis in this article is on giving a comprehensive list of concepts which have been introduced to generalize first countability, from different points of view, along with examples and references. The reader may often judge the value of a concept by considering the number of references pertaining to it, as listed in our references in Section 1.The structure of this survey is as follows: In Section 1, we present a list of definitions of concepts which generalize first countability. For standard terminology we follow Kelley [199], Nagata [290], and Thron [379], and the reader may note that there are some remarks concerning notation and terminology at the beginning of Section 1. The reader is advised to skim Section 1 and refer to it as needed as he would a dictionary. The history of the subject is briefly surveyed in Section 2, along with some motivation for the concepts and some

No abstract

Abstract.A space X is said to be oMK provided that X has a countable closed cover S of metrizable subspaces such that if A-is a compact subset of X, there is a C E 6 for which K C C. A Hausdorff space is o M K and Fréchet if and only if it is representable as a closed image of a metric space obtained by identifying a discrete collection of closed sets with hemicompact boundaries to points.A familiar example of a nonmetrizable space is R/N, that is, the space obtained by identifying the set of natural numbers A, in the set of real numbers R, to a point and giving the resulting set the quotient topology. In [5], the concept of a oMK space proved useful in characterizing certain countably infinite spaces. This note relates identification spaces such as R/N with the concept of a oMK space.All spaces in this paper are understood to be Hausdorff topological spaces and all mappings are continuous onto functions. A space X is oMK provided that X has a countable closed cover G of metrizable subspaces such that if K is a compact subset of X, there is a C E G for which K C C. We may assume that G consists of sets Cx C C2 C • • •, and we will henceforth do so. A space X is Fréchet [2] provided that every accumulation point of a set A in A" is the limit of some sequence in A. It is clear that oMK and Fréchet are each hereditary properties.Theorem I. If a space X is oMK and Fréchet, then it is an image of a metric space M under a closed mapping f, and there is a discrete collection 9 of closed subsets of M, such that f(F) is a point for each F E if, BdyFis hemicompact for each F E % and f is one-to-one upon restriction to M -\J9.The proof follows from a number of propositions. The concept of a oMK space arose in analogy to the concept of hemicompactness introduced by Arens [1]. A space X is hemicompact provided that there is a countable cover G of compact subspaces such that if AT is a compact subset of A, there is a C E G for which K C C. A metrizable space is hemicompact if and only if it is separable and locally compact. Proposition 2. (a) If a space X is oMK and Fréchet, then it is a closed image of a metric space having cardinality that of X.

Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.